This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers’ equations, including the modified Burgers’ equation. The KdV–Burgers’ equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space.

Cognitive Radio Networks (CRNs) have been conceived to improve the efficiency of accessing the spectrum. However, these networks are prone to various kinds of attacks and failures that can compromise the security and performance of their users. One of the notable malicious attacks in cognitive radio networks is the Primary User Emulation (PUE) attack, which results in underutilization and unavailability of the spectrum and low operational efficiency of the network. This study developed an improved technique for detecting PUE attacks in cognitive radio networks and further addressed the characteristics of sparsely populated cognitive radio networks and the mobility of the primary users. A hybrid signal processing-based model was developed using the free space path loss and additive Gaussian noise models. The free space path loss model was used to detect the position of the transmitter, while the additive Gaussian noise model was used to analyze the signal transmitted, i.e., energy detection in the spectrum at the detected location. The proposed model was benchmarked with an existing model using the number of secondary users and the velocity of the transmitter as performance parameters. The simulation results show that the proposed model has improved accuracy in detecting primary user emulation attacks. It was concluded that the proposed hybrid model with respect to the number of secondary users and the velocity of the transmitter can be used for primary user emulation attack detection in cognitive radio networks.

In this paper, we consider the solution of the space fractional diffusion equation using orthogonal collocation on finite elements (OCFE) with quadratic B-spline basis functions. The main advantage of quadratic B-splines is that they have good interpolating properties and can be easily adapted for solving problems on non-uniform grids. The method is unconditionally stable and its convergence is also discussed. It is of order (3 − α) for 1 < α < 2. We present various linear and nonlinear examples. The solutions compared favourably with previous results in the literature.

This paper presents a Simple Generalized Construction of Resolvable Balanced Incomplete Block Designs whose parameter combination is of the form $v=k^2, ~r=k+1, ~\lambda=k^0=1$, where $k$ is prime. The design construction was achieved by using the cyclic subgroup of the symmetric group $S_k$ whose generator is one of the permutations of the $2$-permutation generating set of the Dihedral group $D_k$ and $2$-permutation generating set of the presentation of $S_k$. The method is efficient, sufficient and also mitigate against the tediousness encountered in other methods of construction when $v$ is large.